Chapter 9.1, Problem 26E

MARKET SHARE OF AUTO MANUFACTURERES In a study of the domestic market share of the three major automobile manufacturers $A$, $B$ and $C$ in a certain country, it was found that their current market shares were $60\%$, $30\%$ and $10\%$, respectively. Furthermore, it was found that of the customers who bought a car manufactured by $A$, $75\%$ would again buy a car manufactured by $A$, $15\%$ would buy a car manufactured by $B$, and $10\%$ would buy a car manufactured by $C$. Of the customers who bought a car manufactured by $B$, $90\%$ would again buy a car manufactured by $B$, whereas $5\%$ each would buy cars manufactured by $A$ and $C$. Finally, of the customers who bought a car manufactured by $C$, $85\%$ would again buy a car manufactured by $C$, $5\%$ would buy a car manufactured by $A$, and $10\%$ would buy a car manufactured by $B$. Assuming that these sentiments reflect the buying habits of customers in the future, determine the market share that will be held by each manufacturer after the next two model years.